I. Introduction Experimental Overview Conventional Theory II. Generalized Mean-Field Theory Breakdown of Landau Theory Generalized Mean-Field Theory III. Soft-Mode Field Theory Effective Action Analysis IV. Summary and Conclusion

Dietrich Belitz + University of Oregon * With T.R. Kirkpatrick, M.T. Mercaldo, J. Rollbühler, S. Sessions, T. Vojta + belitz@physics.uoregon.edu

I. INTRODUCTION

1. Experimental Overview

Itinerant ferromagnets whose T_c can be tuned to zero include,

	UGe2, ZrZn2, (MnSi)		(pressure tuned, clean)
	URu2-xRexSi2		(concentration tuned, disordered)
	NixPd1-x		(concentration tuned, clean?)

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Clean materials all show tricritical point , with 2nd order transition at high T, and 1st order transition at low T:

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	MnSi	UGe2	ZrZn2
	( Pfleiderer et al 1997 )	( Pfleiderer & Huxley 2002 )	( Uhlarz et al 2004 )

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T=0 1st order transition persists at nonzero magnetic field, ends at quantum critical point

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Schematic phase diagram:

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This behavior of clean systems appears to be universal

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URuReSi shows 2nd order transition with non-mean-field exponents (e.g., = 3/2) ( Bauer et al 2002 )

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NiPd shows 2nd order transition with mean-field exponents ( Nicklas et al 1999 )

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MnSi shows NFL behavior in PM phase ( Pfleiderer et al 2001 , 2004 )

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2. Conventional Theory

Consider Landau theory with order parameter m = average magnetization

(Free) energy density:

Equation of state:

Landau theory predicts

continuous transition at t=0 with mean-field critical exponents if u > 0,

first order transition if u < 0.

Sandeman et al 2003 , Shick et al 2004 Band structure in UGe₂ leads to u < 0
Origin of 1st order transition at T = 0

Caveats:	Not universal
	TCP needs to be put in by hand

Hertz 1976 Mean-field theory correctly describes T = 0 transition for all d > 1 in clean systems,
and for all d > 0 in disordered ones.

II. GENERALIZED MEAN-FIELD THEORY

1. Breakdown of Landau Theory

Hertz's conclusion is now known to be incorrect: Soft modes in addition to OP fluctuations

Breakdown of Landau theory for d 3 (clean), and d 4 (dirty), respectively
( TRK & DB 1996 ; TRK, DB, et al 1997 ff )

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Phase transition physics is determined by all of the soft modes in the problem.

These are,

OP fluctuations (at t 0, any T)

particle-hole excitations (at T=0, any t)

p-h excitations contribute to

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FM phase broken symmetry
some p-h excitations acquire a mass
contribution to f:

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Note:	particle-hole excitations soft for all values of t
	They are the Goldstone modes of a spontaneously broken symmetry ( Wegner 1979 )
	Generic Scale Invariance
	It is impossible to construct a local field theory in terms of the OP only. (Breakdown of LGW paradigm)
	This is not revolutionary. Analogous effects occur at other phase transitions, Nematic-to-smectic-A in liquid crystals ( Chen, Lubensky, Nelson 1978 ) BCS superconductors (academic case) ( Halperin, Lubensky, Ma 1974 ) Critical dynamics in classical fluids ( Kawasaki 1976 , Halperin & Hohenberg 1977 ) and in particle physics, Fermi theory of weak interactions

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2. Generalized Mean-Field Theory ( DB, TRK, TV 1999 )

Conclusion: Landau theory misses mode-mode coupling contribution to the equation of state:

Generalized mean-field equation of state (d=3):

v > 0 first order transition !

This is generic!

Effects of nonzero temperature (T) and disorder (G) :

T > 0 gives p-h exitations a mass ln m -> ln (m+T) tricritical point

G > 0 changes fermion dispersion relation md -> md/2 G > 0 changes sign of coefficient Generalized mean-field equation of state (d=3) transition second order with non-mean field (and non-classical) exponents etc.

Phase diagrams:

Effect of nonzero magnetic field (h):

Generalized mean-field theory produces the experimentally observed phase diagram NB: So does ordinary Landau theory with a TCP! ( Griffiths 1970 , not well known)

III. SOFT-MODE FIELD THEORY (DB et al 2001a , 2001b , TRK & DB 2002 )